Convex strategies for trajectory optimisation: application to the Polytope Traversal Problem

TL;DR

This paper introduces a convex optimization approach for trajectory planning that efficiently computes feasible paths and time allocations, especially for unstable systems, outperforming traditional methods in speed and success rate.

Contribution

It presents a novel convex formulation for trajectory optimization that guarantees convergence and effectively handles unstable dynamical systems.

Findings

Achieves over 80% success rate in complex polytope traversal scenarios.

Computes feasible trajectories in less than 10ms for simple cases.

Provides initial guesses for nonlinear solvers, improving their convergence.

Abstract

Non-linear Trajectory Optimisation (TO) methods require good initial guesses to converge to a locally optimal solution. A feasible guess can often be obtained by allocating a large amount of time for the trajectory to complete. However for unstable dynamical systems such as humanoid robots, this quasi-static assumption does not always hold. We propose a conservative formulation of the TO problem that simultaneously computes a feasible path and its time allocation. The problem is solved as an efficient convex optimisation problem guaranteed to converge to a locally optimal solution. The interest of the approach is illustrated with the computation of feasible trajectories that traverse sequentially a sequence of polytopes. We demonstrate that on instances of the problem where the quasi static solutions are not admissible, our approach is able to find a feasible solution with a success rate above $80 \%$ in all the scenarios considered, in less than 10ms for problems involving traversing less than 5 polytopes and less than 1s for problems involving 20 polytopes, thus demonstrating its ability to reliably provide initial guesses to advanced non linear solvers.